\(\int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} (-b+a x^4+2 x^8)} \, dx\) [1896]

Optimal result

Integrand size = 35, antiderivative size = 131 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=-\frac {\text {RootSum}\left [2 a^2-2 b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)-a b \log (x)+2 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+a b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(491\) vs. \(2(131)=262\).

Time = 0.80 (sec) , antiderivative size = 491, normalized size of antiderivative = 3.75, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6860, 385, 218, 214, 211} \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\frac {\left (a-\frac {a^2+8}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-4 b}}+\frac {\left (\frac {a^2+8}{\sqrt {a^2+8 b}}+a\right ) \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-4 b}}+\frac {\left (a-\frac {a^2+8}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2-4 b}}+\frac {\left (\frac {a^2+8}{\sqrt {a^2+8 b}}+a\right ) \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2-4 b}} \]

[In]

Int[(-2 + a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + 2*x^8)),x]

[Out]

((a - (8 + a^2)/Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 4*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)
*(b + a*x^4)^(1/4))])/(2*(a - Sqrt[a^2 + 8*b])^(3/4)*(a^2 - 4*b - a*Sqrt[a^2 + 8*b])^(1/4)) + ((a + (8 + a^2)/
Sqrt[a^2 + 8*b])*ArcTan[((a^2 - 4*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/
4))])/(2*(a + Sqrt[a^2 + 8*b])^(3/4)*(a^2 - 4*b + a*Sqrt[a^2 + 8*b])^(1/4)) + ((a - (8 + a^2)/Sqrt[a^2 + 8*b])
*ArcTanh[((a^2 - 4*b - a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a - Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a - S
qrt[a^2 + 8*b])^(3/4)*(a^2 - 4*b - a*Sqrt[a^2 + 8*b])^(1/4)) + ((a + (8 + a^2)/Sqrt[a^2 + 8*b])*ArcTanh[((a^2
- 4*b + a*Sqrt[a^2 + 8*b])^(1/4)*x)/((a + Sqrt[a^2 + 8*b])^(1/4)*(b + a*x^4)^(1/4))])/(2*(a + Sqrt[a^2 + 8*b])
^(3/4)*(a^2 - 4*b + a*Sqrt[a^2 + 8*b])^(1/4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+\frac {-8-a^2}{\sqrt {a^2+8 b}}}{\left (a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {a-\frac {-8-a^2}{\sqrt {a^2+8 b}}}{\left (a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx \\ & = \left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx \\ & = \left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a^2+8 b}-\left (-4 b+a \left (a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a^2+8 b}-\left (-4 b+a \left (a+\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2-4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}+\frac {\left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2-4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}+\frac {\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2-4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}+\frac {\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2-4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}} \\ & = \frac {\left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-4 b-a \sqrt {a^2+8 b}}}+\frac {\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-4 b+a \sqrt {a^2+8 b}}}+\frac {\left (a-\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-4 b-a \sqrt {a^2+8 b}}}+\frac {\left (a+\frac {8+a^2}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2-4 b+a \sqrt {a^2+8 b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=-\frac {\text {RootSum}\left [2 a^2-2 b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)+a b \log (x)-2 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-a b \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]

[In]

Integrate[(-2 + a*x^4)/((b + a*x^4)^(1/4)*(-b + a*x^4 + 2*x^8)),x]

[Out]

-1/4*RootSum[2*a^2 - 2*b - 3*a*#1^4 + #1^8 & , (2*a*Log[x] + a*b*Log[x] - 2*a*Log[(b + a*x^4)^(1/4) - x*#1] -
a*b*Log[(b + a*x^4)^(1/4) - x*#1] - 2*Log[x]*#1^4 + 2*Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1 + 2*#1^5) &
]/b

Maple [N/A] (verified)

Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.54

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-2 b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{4}+a \left (2+b \right )\right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{5}+3 \textit {\_R} a}}{4 b}\)	\(71\)

[In]

int((a*x^4-2)/(a*x^4+b)^(1/4)/(2*x^8+a*x^4-b),x,method=_RETURNVERBOSE)

[Out]

-1/4*sum((-2*_R^4+a*(2+b))*ln((-_R*x+(a*x^4+b)^(1/4))/x)/(-2*_R^5+3*_R*a),_R=RootOf(_Z^8-3*_Z^4*a+2*a^2-2*b))/
b

Fricas [F(-1)]

Timed out. \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x^4-2)/(a*x^4+b)^(1/4)/(2*x^8+a*x^4-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 58.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} - 2}{\sqrt [4]{a x^{4} + b} \left (a x^{4} - b + 2 x^{8}\right )}\, dx \]

[In]

integrate((a*x**4-2)/(a*x**4+b)**(1/4)/(2*x**8+a*x**4-b),x)

[Out]

Integral((a*x**4 - 2)/((a*x**4 + b)**(1/4)*(a*x**4 - b + 2*x**8)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.27 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2}{{\left (2 \, x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*x^4-2)/(a*x^4+b)^(1/4)/(2*x^8+a*x^4-b),x, algorithm="maxima")

[Out]

integrate((a*x^4 - 2)/((2*x^8 + a*x^4 - b)*(a*x^4 + b)^(1/4)), x)

Giac [N/A]

Not integrable

Time = 1.67 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.27 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2}{{\left (2 \, x^{8} + a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*x^4-2)/(a*x^4+b)^(1/4)/(2*x^8+a*x^4-b),x, algorithm="giac")

[Out]

integrate((a*x^4 - 2)/((2*x^8 + a*x^4 - b)*(a*x^4 + b)^(1/4)), x)

Mupad [N/A]

Not integrable

Time = 5.58 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.27 \[ \int \frac {-2+a x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^4+2 x^8\right )} \, dx=\int \frac {a\,x^4-2}{{\left (a\,x^4+b\right )}^{1/4}\,\left (2\,x^8+a\,x^4-b\right )} \,d x \]

[In]

int((a*x^4 - 2)/((b + a*x^4)^(1/4)*(a*x^4 - b + 2*x^8)),x)

[Out]

int((a*x^4 - 2)/((b + a*x^4)^(1/4)*(a*x^4 - b + 2*x^8)), x)